algebra precalculus – Solutions for $ 2 ^ x + 2 ^ y = 2 ^ z $ for integers x, y and z

I'm trying to prove / disprove that the equation

$ 2 ^ x + 2 ^ y = 2 ^ z $

or $ x $, $ y $ and $ z $ are positive integers, only have trivial solutions. The obvious case where this is true would be for $ x = y $ but I'm not sure if solutions exist for $ x only $

I have the impression that there are no other solutions, but I do not know how one could officially prove it. Someone has advice on how to do this?

Sorry it's about a trivial issue / a known problem.